III. Angular Momentum Conservation (Chap. 10) Rotation. We repeat Chap. 28 with rotatiing objects. Eqs. of motion. Energy.


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1 Chap. 10: Rotational Motion I. Rotational Kinematics II. Rotational Dynamics  Newton s Law for Rotation III. Angular Momentum Conservation (Chap. 10) 1 Toward Exam 3 Eqs. of motion o To study angular velocity ( ) and angular acceleration ( ). o To examine rotation with constant angular acceleration. o To understand the relationship between linear and angular quantities. Energy o To determine the kinetic energy of rotation (K rot = ½ I 2 ) and the moment of inertia (I). o To study rotation about a moving axis. More o Force (F = m a) Torque ( = I ) o Work (W) o Power (P) o Momentum (p) Angular Momentum (L) We repeat Chap. 28 with rotatiing objects. 2
2 Describing the rotation Rotational Motion 3 Race of the objects on a ramp Why? Phy201 Problem 4
3 Think of a rolling object FBD Newton s 2 nd law 5 Torque and Moment of Inertia a x m (b) F (a) F I Speeding up l a l b Mass (kg) is a measure of the inertia of a body. How difficult to move an object. FBD Newton s 2 nd law 6
4 Q10.7 A solid bowling ball rolls down a ramp. Which of the following forces exerts a torque on the bowling ball about its center? A. the weight of the ball B. the normal force exerted by the ramp C. the friction force exerted by the ramp D. more than one of the above E. The answer depends on whether the ball rolls without slipping. 7 A10.7 A solid bowling ball rolls down a ramp. Which of the following forces exerts a torque on the bowling ball about its center? A. the weight of the ball B. the normal force exerted by the ramp C. the friction force exerted by the ramp D. more than one of the above E. The answer depends on whether the ball rolls without slipping. 8
5 FBD for Rotational Motion Analyze the rolling sphere in terms of forces and torques: find the magnitudes of the velocity v and. Rotational Motion 9 Intentionally left with blank page 10
6 h m = 3.00 kg = 60.0 o Revisiting Example 3 l = 5.00 m No friction h Solid disk m = 3.00 kg r = m Rolling without slipping (with friction) s = Example 3 Worksheet 12
7 Speed of YoYo A sting is wrapped around a uniform solid cylinder of mass M (= kg) and radius R (= m), and the cylinder starts falling from rest. (a) Draw the freebody diagram for the cylinder while it descends. (b) Find its acceleration and the tension in the string. (c) Find the speed after the cylinder has descended h = m. 13 Forces on YoYo F.B.D. 14
8 Chap. 10: Rotational Motion I. Rotational Kinematics II. Rotational Dynamics  Newton s Law for Rotation III. Angular Momentum Conservation 15 What We Learned with Example 3 m = 3.00 kg No friction h l = 5.00 m h = 60.0 o Solid disk m = 3.00 kg r = m v f = 9.21 m/s Rolling without slipping (with friction) s = FBD Newton s 2 nd law for rotation (torque) = (moment of inertia) x v f = 7.50 m/s a = 5.66 m/s 2 16
9 Example 1 Was: FBD Newton s 2 nd law Momentum conservation Now: FBD Newton s 2 nd law for rotation Angular momentum conservation 17 Anatomy of Example 1 Angular momentum conservation WE theorem F rad = m a rad 18
10 Angular Momentum Conservation (I) P Rotational Motion 19 Angular Momentum Conservation (II) Rotational Motion 20
11 Which one has a bigger I? 21 Which one has a bigger I? 22
12 Make Sense? 2 (d) A door (solid rectangular thin plate) of mass M = 15.0 kg is free to rotate on about hinge line. Calculate the moment of inertia of the door about the hinge line. H W 23 M rim = 11 kg R out = m M spoke = 3.5 kg R in = m 24
13 Angular Momentum Conservation I a : big I b : small z I a : big I b : small z The gravitational force and normal force don t do anything about the rotation. The net torque is zero. We observe the rotational speed increases. angular velocity ( omega ) increases. Why? 25 Angular Momentum Conservation I a : big I b : small z I a : big I b : small z The gravitational force and normal force don t do anything about the rotation. The net torque is zero. 26
14 Example 2 I i =MR 2 i m 1 I f =MR f 2 m 2 f R f m 1 i i f f 27 Example 2 Workbook 28
15 Example 3 A person stands, hands at the side, on a platform that is rotating at a rate of 1.60 rev/s. If the person now raises his arms to a horizontal position, the speed of rotation decreases to rev/s. (a) By what factor has the moment of inertia of the person changed? (b) Compare K i and K f i i f f 29 Example 3 Workbook 30
16 Example 4 (at Home) A professor E=mc 2 stands at the center of a turntable, holding his arms extended horizontally, with a 5.00 kg dumbbell in each hand. He is set rotating about a vertical axis, making one revolution in 2.00 sec. The professor with outstretched hands and arms can be considered a solid cylinder (mass M = 50.0 kg and height H = 1.50 m, radius R = m) plus very light rods of length l = m and particle mass m = 5.00 kg, which is attached to the solid cylinder. See figure A. When his hands and arms are brought in near his body, it can be modeled as in figure B. Find the magnitude of the final angular velocity of the professor. Make sure you show the steps to find the final numerical answer. 31 Example 4 Workbook 32
17 Analogy 33 Intentionally left with blank page 34
18 Chap. 10: Static Equilibrium 35 Real Life Phys 201 Static Equilibrium 36
19 Real Life Phys 201 Static Equilibrium 37 Static Problems Static No translational motion (a = 0) & No rotational motion ( = 0) (1) F x = m a x = 0 (2) F y = m a y = 0 (3) = I = 0 38
20 Finding R R Rotational Motion 39 Intentionally left with blank page 40
21 Example 5 A uniform drawbridge must be held at a 37.0 o angle above the horizontal to allow ships to pass underneath. The drawbridge weighs Mg = N is L = 14.0 m long, and pivots about a hinge at its lower end. A cable is connected 3.50 m from the hinge, as measured along the bridge, and pulls horizontally on the bridge to hold it in place. (a) What is the tension T in the cable? (b) Find the magnitude of the force the hinge exerts on the bridge. (c) Find the direction of the force the hinge exerts on the bridge. (d) If the cable suddenly breaks, what is the initial angular acceleration of the bridge? 41 Example 5 Workbook (I) 42
22 Example 5 Workbook (II) 43 Intentionally left with blank page 44
23 Example 6 (at Home) T =? F V =? M = 1500 kg L = 8.00 m F H =? Static no translational motion & no rotational motion 45 Example 6 Workbook FBD Three Equations Solve a system of equations [Try] 1042, 1043, 1046, 1047, Plus 101,
24 Try this! Static Equilibrium 47 Try this! Static Equilibrium 48
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